Likelihood function of gamma distribution

Let X1, X2,...Xn be i.i.d. Gamma(a,b) random variables where both a and b are unknown. Let T1= (X1 + X2 +...+ Xn)/n and T2 = (X21 + X22 +...+ X2n)/n be the first and second sample moments, respectively. Show the likelihood function of a and b.

So...I don't really understand what a likelihood function is. I believe it is the same algebraically as f(x1|a,b)*f(x2 | a,b)*...*f(xn | a,b), but I don't really know how to find this in terms of a and b. Any help is much appreciated. Thanks so much!

Re: Likelihood function of gamma distribution

What I was thinking was that the likelihood function is the (pdf of gamma)^n. So, the probability function is: f(x1,x2,...xn|a,b) = [(b^a)/gamma(a)]*x^(a-1)e^(-bx). Then, I would get the likelihood function to be L(a,b | x1, x2,...,xn)= f(x1,x2,...xn|a,b) ^n where x1...xn are represented in summation form. Is this the right direction to take it? I also don't really know what the support of that function might be. I think a and b are > 0?